# Smith chart

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The Smith chart, invented by Phillip H. Smith (1905–1987), [1] [2] is a graphical aid or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assist in solving problems with transmission lines and matching circuits. [3] The Smith chart can be used to simultaneously display multiple parameters including impedances, admittances, reflection coefficients, ${\displaystyle S_{nn}\,}$ scattering parameters, noise figure circles, constant gain contours and regions for unconditional stability, including mechanical vibrations analysis. [4] [5] The Smith chart is most frequently used at or within the unity radius region. However, the remainder is still mathematically relevant, being used, for example, in oscillator design and stability analysis. [6]

Phillip Hagar Smith was an electrical engineer, who became famous for his invention of the Smith chart. Smith graduated from Tufts College in 1928 with a BS degree in electrical engineering. While working for Bell Telephone Laboratories, he invented his eponymous Smith chart.

A nomogram, also called a nomograph, alignment chart or abaque, is a graphical calculating device, a two-dimensional diagram designed to allow the approximate graphical computation of a mathematical function. The field of nomography was invented in 1884 by the French engineer Philbert Maurice d’Ocagne (1862-1938) and used extensively for many years to provide engineers with fast graphical calculations of complicated formulas to a practical precision. Nomograms use a parallel coordinate system invented by d'Ocagne rather than standard Cartesian coordinates.

Electrical engineering is a professional engineering discipline that generally deals with the study and application of electricity, electronics, and electromagnetism. This field first became an identifiable occupation in the later half of the 19th century after commercialization of the electric telegraph, the telephone, and electric power distribution and use. Subsequently, broadcasting and recording media made electronics part of daily life. The invention of the transistor, and later the integrated circuit, brought down the cost of electronics to the point they can be used in almost any household object.

## Contents

While the use of paper Smith charts for solving the complex mathematics involved in matching problems has been largely replaced by software based methods, the Smith chart display is still the preferred method of displaying how RF parameters behave at one or more frequencies, an alternative to using tabular information. Thus most RF circuit analysis software includes a Smith chart option for the display of results and all but the simplest impedance measuring instruments can display measured results on a Smith chart display.(depending on the question(s).)

A table is an arrangement of data in rows and columns, or possibly in a more complex structure. Tables are widely used in communication, research, and data analysis. Tables appear in print media, handwritten notes, computer software, architectural ornamentation, traffic signs, and many other places. The precise conventions and terminology for describing tables vary depending on the context. Further, tables differ significantly in variety, structure, flexibility, notation, representation and use. In books and technical articles, tables are typically presented apart from the main text in numbered and captioned floating blocks.

## Overview

The Smith chart is plotted on the complex reflection coefficient plane in two dimensions and is scaled in normalised impedance (the most common), normalised admittance or both, using different colours to distinguish between them. These are often known as the Z, Y and YZ Smith charts respectively. [7] Normalised scaling allows the Smith chart to be used for problems involving any characteristic or system impedance which is represented by the center point of the chart. The most commonly used normalization impedance is 50 ohms. Once an answer is obtained through the graphical constructions described below, it is straightforward to convert between normalised impedance (or normalised admittance) and the corresponding unnormalized value by multiplying by the characteristic impedance (admittance). Reflection coefficients can be read directly from the chart as they are unitless parameters.

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, and b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.

In physics and electrical engineering the reflection coefficient is a parameter that describes how much of an electromagnetic wave is reflected by an impedance discontinuity in the transmission medium. It is equal to the ratio of the amplitude of the reflected wave to the incident wave, with each expressed as phasors. For example, it is used in optics to calculate the amount of light that is reflected from a surface with a different index of refraction, such as a glass surface, or in an electrical transmission line to calculate how much of the electromagnetic wave is reflected by an impedance. The reflection coefficient is closely related to the transmission coefficient. The reflectance of a system is also sometimes called a "reflection coefficient".

In physics and mathematics, the dimension of a mathematical space is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces.

The Smith chart has circumferential scaling in wavelengths and degrees. The wavelengths scale is used in distributed component problems and represents the distance measured along the transmission line connected between the generator or source and the load to the point under consideration. The degrees scale represents the angle of the voltage reflection coefficient at that point. The Smith chart may also be used for lumped element matching and analysis problems.

In geometry, the circumference of a circle is the (linear) distance around it. That is, the circumference would be the length of the circle if it were opened up and straightened out to a line segment. Since a circle is the edge (boundary) of a disk, circumference is a special case of perimeter. The perimeter is the length around any closed figure and is the term used for most figures excepting the circle and some circular-like figures such as ellipses. Informally, "circumference" may also refer to the edge itself rather than to the length of the edge.

A degree, usually denoted by °, is a measurement of a plane angle, defined so that a full rotation is 360 degrees.

In electrical engineering, the distributed element model or transmission line model of electrical circuits assumes that the attributes of the circuit are distributed continuously throughout the material of the circuit. This is in contrast to the more common lumped element model, which assumes that these values are lumped into electrical components that are joined by perfectly conducting wires. In the distributed element model, each circuit element is infinitesimally small, and the wires connecting elements are not assumed to be perfect conductors; that is, they have impedance. Unlike the lumped element model, it assumes non-uniform current along each branch and non-uniform voltage along each node. The distributed model is used at high frequencies where the wavelength becomes comparable to the physical dimensions of the circuit, making the lumped model inaccurate.

Use of the Smith chart and the interpretation of the results obtained using it requires a good understanding of AC circuit theory and transmission line theory, both of which are pre-requisites for RF engineers.

Alternating current (AC) is an electric current which periodically reverses direction, in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in which electric power is delivered to businesses and residences, and it is the form of electrical energy that consumers typically use when they plug kitchen appliances, televisions, fans and electric lamps into a wall socket. A common source of DC power is a battery cell in a flashlight. The abbreviations AC and DC are often used to mean simply alternating and direct, as when they modify current or voltage.

As impedances and admittances change with frequency, problems using the Smith chart can only be solved manually using one frequency at a time, the result being represented by a point. This is often adequate for narrow band applications (typically up to about 5% to 10% bandwidth) but for wider bandwidths it is usually necessary to apply Smith chart techniques at more than one frequency across the operating frequency band. Provided the frequencies are sufficiently close, the resulting Smith chart points may be joined by straight lines to create a locus.

Frequency is the number of occurrences of a repeating event per unit of time. It is also referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency. The period is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example: if a newborn baby's heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals (sound), radio waves, and light.

In modern mathematics, a point refers usually to an element of some set called a space.

Bandwidth is the difference between the upper and lower frequencies in a continuous band of frequencies. It is typically measured in hertz, and depending on context, may specifically refer to passband bandwidth or baseband bandwidth. Passband bandwidth is the difference between the upper and lower cutoff frequencies of, for example, a band-pass filter, a communication channel, or a signal spectrum. Baseband bandwidth applies to a low-pass filter or baseband signal; the bandwidth is equal to its upper cutoff frequency.

A locus of points on a Smith chart covering a range of frequencies can be used to visually represent:

• how capacitive or how inductive a load is across the frequency range
• how difficult matching is likely to be at various frequencies
• how well matched a particular component is.

The accuracy of the Smith chart is reduced for problems involving a large locus of impedances or admittances, although the scaling can be magnified for individual areas to accommodate these.

## Mathematical basis

### Actual and normalised impedance and admittance

A transmission line with a characteristic impedance of ${\displaystyle Z_{0}\,}$ may be universally considered to have a characteristic admittance of ${\displaystyle Y_{0}\,}$ where

${\displaystyle Y_{0}={\frac {1}{Z_{0}}}\,}$

Any impedance, ${\displaystyle Z_{\text{T}}\,}$ expressed in ohms, may be normalised by dividing it by the characteristic impedance, so the normalised impedance using the lower case zT is given by

${\displaystyle z_{\text{T}}={\frac {Z_{\text{T}}}{Z_{0}}}\,}$

Similarly, for normalised admittance

${\displaystyle y_{\text{T}}={\frac {Y_{\text{T}}}{Y_{0}}}\,}$

The SI unit of impedance is the ohm with the symbol of the upper case Greek letter omega (Ω) and the SI unit for admittance is the siemens with the symbol of an upper case letter S. Normalised impedance and normalised admittance are dimensionless. Actual impedances and admittances must be normalised before using them on a Smith chart. Once the result is obtained it may be de-normalised to obtain the actual result.

### The normalised impedance Smith chart

Using transmission line theory, if a transmission line is terminated in an impedance (${\displaystyle Z_{\text{T}}\,}$) which differs from its characteristic impedance (${\displaystyle Z_{0}\,}$), a standing wave will be formed on the line comprising the resultant of both the incident or forward (${\displaystyle V_{\text{F}}\,}$) and the reflected or reversed (${\displaystyle V_{\text{R}}\,}$) waves. Using complex exponential notation:

${\displaystyle V_{\text{F}}=A\exp(j\omega t)\exp(+\gamma \ell )~\,}$ and
${\displaystyle V_{\text{R}}=B\exp(j\omega t)\exp(-\gamma \ell )\,}$

where

${\displaystyle \exp(j\omega t)\,}$ is the temporal part of the wave
${\displaystyle \exp(\pm \gamma \ell )\,}$ is the spatial part of the wave and
${\displaystyle \omega =2\pi f\,}$ where
${\displaystyle \omega \,}$ is the angular frequency in radians per second (rad/s)
${\displaystyle f\,}$ is the frequency in hertz (Hz)
${\displaystyle t\,}$ is the time in seconds (s)
${\displaystyle A\,}$ and ${\displaystyle B\,}$ are constants
${\displaystyle \ell \,}$ is the distance measured along the transmission line from the load toward the generator in metres (m)

Also

${\displaystyle \gamma =\alpha +j\beta \,}$ is the propagation constant which has units 1/m

where

${\displaystyle \alpha \,}$ is the attenuation constant in nepers per metre (Np/m)
${\displaystyle \beta \,}$ is the phase constant in radians per metre (rad/m)

The Smith chart is used with one frequency (${\displaystyle \omega }$) at a time, and only for one moment (${\displaystyle t}$) at a time, so the temporal part of the phase (${\displaystyle \exp(j\omega t)\,}$) is fixed. All terms are actually multiplied by this to obtain the instantaneous phase, but it is conventional and understood to omit it. Therefore,

${\displaystyle V_{\text{F}}=A\exp(+\gamma \ell )\,}$ and
${\displaystyle V_{\text{R}}=B\exp(-\gamma \ell )\,}$

where ${\displaystyle A\,}$ and ${\displaystyle B\,}$ are respectively the forward and reverse voltage amplitudes at the load.

#### The variation of complex reflection coefficient with position along the line

The complex voltage reflection coefficient ${\displaystyle \Gamma \,}$ is defined as the ratio of the reflected wave to the incident (or forward) wave. Therefore,

${\displaystyle \Gamma ={\frac {V_{\text{R}}}{V_{\text{F}}}}={\frac {B\exp(-\gamma \ell )}{A\exp(+\gamma \ell )}}=C\exp(-2\gamma \ell )\,}$

where C is also a constant.

For a uniform transmission line (in which ${\displaystyle \gamma \,}$ is constant), the complex reflection coefficient of a standing wave varies according to the position on the line. If the line is lossy (${\displaystyle \alpha \,}$ is non-zero) this is represented on the Smith chart by a spiral path. In most Smith chart problems however, losses can be assumed negligible (${\displaystyle \alpha =0\,}$) and the task of solving them is greatly simplified. For the loss free case therefore, the expression for complex reflection coefficient becomes

${\displaystyle \Gamma =\Gamma _{\text{L}}\exp(-2j\beta \ell )\,}$

where ${\displaystyle \Gamma _{\text{L}}\,}$ is the reflection coefficient at the load, and ${\displaystyle \ell \,}$ is the line length from the load to the location where the reflection coefficient is measured. The phase constant ${\displaystyle \beta \,}$ may also be written as

${\displaystyle \beta ={\frac {2\pi }{\lambda }}\,}$

where ${\displaystyle \lambda \,}$ is the wavelength within the transmission line at the test frequency.

Therefore,

${\displaystyle \Gamma =\Gamma _{\text{L}}\exp \left({\frac {-4j\pi }{\lambda }}\ell \right)\,}$

This equation shows that, for a standing wave, the complex reflection coefficient and impedance repeats every half wavelength along the transmission line. The complex reflection coefficient is generally simply referred to as reflection coefficient. The outer circumferential scale of the Smith chart represents the distance from the generator to the load scaled in wavelengths and is therefore scaled from zero to 0.50 .

#### The variation of normalised impedance with position along the line

If ${\displaystyle V\,}$ and ${\displaystyle I\,}$ are the voltage across and the current entering the termination at the end of the transmission line respectively, then

${\displaystyle V_{\text{F}}+V_{\text{R}}=V\,}$ and
${\displaystyle V_{\text{F}}-V_{\text{R}}=Z_{0}I\,}$.

By dividing these equations and substituting for both the voltage reflection coefficient

${\displaystyle \Gamma ={\frac {V_{\text{R}}}{V_{\text{F}}}}\,}$

and the normalised impedance of the termination represented by the lower case z, subscript T

${\displaystyle z_{\text{T}}={\frac {V}{Z_{0}I}}\,}$

gives the result:

${\displaystyle z_{\text{T}}={\frac {1+\Gamma }{1-\Gamma }}\,}$.

Alternatively, in terms of the reflection coefficient

${\displaystyle \Gamma ={\frac {z_{\text{T}}-1}{z_{\text{T}}+1}}\,}$

These are the equations which are used to construct the Z Smith chart. Mathematically speaking ${\displaystyle \Gamma \,}$ and ${\displaystyle z_{\text{T}}\,}$ are related via a Möbius transformation.

Both ${\displaystyle \Gamma \,}$ and ${\displaystyle z_{\text{T}}\,}$ are expressed in complex numbers without any units. They both change with frequency so for any particular measurement, the frequency at which it was performed must be stated together with the characteristic impedance.

${\displaystyle \Gamma \,}$ may be expressed in magnitude and angle on a polar diagram. Any actual reflection coefficient must have a magnitude of less than or equal to unity so, at the test frequency, this may be expressed by a point inside a circle of unity radius. The Smith chart is actually constructed on such a polar diagram. The Smith chart scaling is designed in such a way that reflection coefficient can be converted to normalised impedance or vice versa. Using the Smith chart, the normalised impedance may be obtained with appreciable accuracy by plotting the point representing the reflection coefficient treating the Smith chart as a polar diagram and then reading its value directly using the characteristic Smith chart scaling. This technique is a graphical alternative to substituting the values in the equations.

By substituting the expression for how reflection coefficient changes along an unmatched loss free transmission line

${\displaystyle \Gamma ={\frac {B\exp(-\gamma \ell )}{A\exp(\gamma \ell )}}={\frac {B\exp(-j\beta \ell )}{A\exp(j\beta \ell )}}\,}$

for the loss free case, into the equation for normalised impedance in terms of reflection coefficient

${\displaystyle z_{\text{T}}={\frac {1+\Gamma }{1-\Gamma }}\,}$.

and using Euler's formula

${\displaystyle \exp(j\theta )=\cos \theta +j\sin \theta \,}$

yields the impedance version transmission line equation for the loss free case: [8]

${\displaystyle Z_{\text{in}}=Z_{0}{\frac {Z_{\text{L}}+jZ_{0}\tan(\beta \ell )}{Z_{0}+jZ_{\text{L}}\tan(\beta \ell )}}\,}$

where ${\displaystyle Z_{\text{in}}\,}$ is the impedance 'seen' at the input of a loss free transmission line of length ${\displaystyle \ell }$, terminated with an impedance ${\displaystyle Z_{\text{L}}\,}$

Versions of the transmission line equation may be similarly derived for the admittance loss free case and for the impedance and admittance lossy cases.

The Smith chart graphical equivalent of using the transmission line equation is to normalise ${\displaystyle Z_{\text{L}}\,}$, to plot the resulting point on a Z Smith chart and to draw a circle through that point centred at the Smith chart centre. The path along the arc of the circle represents how the impedance changes whilst moving along the transmission line. In this case the circumferential (wavelength) scaling must be used, remembering that this is the wavelength within the transmission line and may differ from the free space wavelength.

#### Regions of the Z Smith chart

If a polar diagram is mapped on to a cartesian coordinate system it is conventional to measure angles relative to the positive x-axis using a counterclockwise direction for positive angles. The magnitude of a complex number is the length of a straight line drawn from the origin to the point representing it. The Smith chart uses the same convention, noting that, in the normalised impedance plane, the positive x-axis extends from the center of the Smith chart at ${\displaystyle z_{\text{T}}=1\pm j0\,}$ to the point ${\displaystyle z_{\text{T}}=\infty \pm j\infty \,}$. The region above the x-axis represents inductive impedances (positive imaginary parts) and the region below the x-axis represents capacitive impedances (negative imaginary parts).

If the termination is perfectly matched, the reflection coefficient will be zero, represented effectively by a circle of zero radius or in fact a point at the centre of the Smith chart. If the termination was a perfect open circuit or short circuit the magnitude of the reflection coefficient would be unity, all power would be reflected and the point would lie at some point on the unity circumference circle.

#### Circles of constant normalised resistance and constant normalised reactance

The normalised impedance Smith chart is composed of two families of circles: circles of constant normalised resistance and circles of constant normalised reactance. In the complex reflection coefficient plane the Smith chart occupies a circle of unity radius centred at the origin. In cartesian coordinates therefore the circle would pass through the points (+1,0) and (1,0) on the x-axis and the points (0,+1) and (0,1) on the y-axis.

Since both ${\displaystyle \Gamma }$ and ${\displaystyle z\,}$ are complex numbers, in general they may be written as:

${\displaystyle z=a+jb\,}$
${\displaystyle \Gamma =c+jd\,}$

with a, b, c and d real numbers.

Substituting these into the equation relating normalised impedance and complex reflection coefficient:

${\displaystyle \Gamma ={\frac {z-1}{z+1}}\,}$

gives the following result:

${\displaystyle \Gamma =c+jd=\left[{\frac {a^{2}+b^{2}-1}{(a+1)^{2}+b^{2}}}\right]+j\left[{\frac {2b}{(a+1)^{2}+b^{2}}}\right]\,}$.

This is the equation which describes how the complex reflection coefficient changes with the normalised impedance and may be used to construct both families of circles. [9]

### The Y Smith chart

The Y Smith chart is constructed in a similar way to the Z Smith chart case but by expressing values of voltage reflection coefficient in terms of normalised admittance instead of normalised impedance. The normalised admittance yT is the reciprocal of the normalised impedance zT, so

${\displaystyle y_{\text{T}}={\frac {1}{z_{\text{T}}}}\,}$

Therefore:

${\displaystyle y_{\text{T}}={\frac {1-\Gamma }{1+\Gamma }}\,}$

and

${\displaystyle \Gamma ={\frac {1-y_{\text{T}}}{1+y_{\text{T}}}}\,}$

The Y Smith chart appears like the normalised impedance type but with the graphic scaling rotated through 180°, the numeric scaling remaining unchanged.

The region above the x-axis represents capacitive admittances and the region below the x-axis represents inductive admittances. Capacitive admittances have positive imaginary parts and inductive admittances have negative imaginary parts.

Again, if the termination is perfectly matched the reflection coefficient will be zero, represented by a 'circle' of zero radius or in fact a point at the centre of the Smith chart. If the termination was a perfect open or short circuit the magnitude of the voltage reflection coefficient would be unity, all power would be reflected and the point would lie at some point on the unity circumference circle of the Smith chart.

### Practical examples

A point with a reflection coefficient magnitude 0.63 and angle 60° represented in polar form as ${\displaystyle 0.63\angle 60^{\circ }\,}$, is shown as point P1 on the Smith chart. To plot this, one may use the circumferential (reflection coefficient) angle scale to find the ${\displaystyle \angle 60^{\circ }\,}$ graduation and a ruler to draw a line passing through this and the centre of the Smith chart. The length of the line would then be scaled to P1 assuming the Smith chart radius to be unity. For example, if the actual radius measured from the paper was 100 mm, the length OP1 would be 63 mm.

The following table gives some similar examples of points which are plotted on the Z Smith chart. For each, the reflection coefficient is given in polar form together with the corresponding normalised impedance in rectangular form. The conversion may be read directly from the Smith chart or by substitution into the equation.

Some examples of points plotted on the normalised impedance Smith chart
Point IdentityReflection Coefficient (Polar Form)Normalised Impedance (Rectangular Form)
P1 (Inductive)${\displaystyle 0.63\angle 60^{\circ }\,}$${\displaystyle 0.80+j1.40\,}$
P2 (Inductive)${\displaystyle 0.73\angle 125^{\circ }\,}$${\displaystyle 0.20+j0.50\,}$
P3 (Capacitive)${\displaystyle 0.44\angle -116^{\circ }\,}$${\displaystyle 0.50-j0.50\,}$

### Working with both the Z Smith chart and the Y Smith charts

In RF circuit and matching problems sometimes it is more convenient to work with admittances (representing conductances and susceptances) and sometimes it is more convenient to work with impedances (representing resistances and reactances). Solving a typical matching problem will often require several changes between both types of Smith chart, using normalised impedance for series elements and normalised admittances for parallel elements. For these a dual (normalised) impedance and admittance Smith chart may be used. Alternatively, one type may be used and the scaling converted to the other when required. In order to change from normalised impedance to normalised admittance or vice versa, the point representing the value of reflection coefficient under consideration is moved through exactly 180 degrees at the same radius. For example, the point P1 in the example representing a reflection coefficient of ${\displaystyle 0.63\angle 60^{\circ }\,}$ has a normalised impedance of ${\displaystyle z_{P}=0.80+j1.40\,}$. To graphically change this to the equivalent normalised admittance point, say Q1, a line is drawn with a ruler from P1 through the Smith chart centre to Q1, an equal radius in the opposite direction. This is equivalent to moving the point through a circular path of exactly 180 degrees. Reading the value from the Smith chart for Q1, remembering that the scaling is now in normalised admittance, gives ${\displaystyle y_{P}=0.30-j0.54\,}$. Performing the calculation

${\displaystyle y_{\text{T}}={\frac {1}{z_{\text{T}}}}\,}$

manually will confirm this.

Once a transformation from impedance to admittance has been performed, the scaling changes to normalised admittance until a later transformation back to normalised impedance is performed.

The table below shows examples of normalised impedances and their equivalent normalised admittances obtained by rotation of the point through 180°. Again, these may be obtained either by calculation or using a Smith chart as shown, converting between the normalised impedance and normalised admittances planes.

Values of reflection coefficient as normalised impedances and the equivalent normalised admittances
Normalised Impedance PlaneNormalised Admittance Plane
P1 (${\displaystyle z=0.80+j1.40\,}$)Q1 (${\displaystyle y=0.30-j0.54\,}$)
P10 (${\displaystyle z=0.10+j0.22\,}$)Q10 (${\displaystyle y=1.80-j3.90\,}$)

### Choice of Smith chart type and component type

The choice of whether to use the Z Smith chart or the Y Smith chart for any particular calculation depends on which is more convenient. Impedances in series and admittances in parallel add while impedances in parallel and admittances in series are related by a reciprocal equation. If ${\displaystyle Z_{\text{TS}}}$ is the equivalent impedance of series impedances and ${\displaystyle Z_{\text{TP}}}$ is the equivalent impedance of parallel impedances, then

${\displaystyle Z_{\text{TS}}=Z_{1}+Z_{2}+Z_{3}+...\,}$
${\displaystyle {\frac {1}{Z_{\text{TP}}}}={\frac {1}{Z_{1}}}+{\frac {1}{Z_{2}}}+{\frac {1}{Z_{3}}}+...\,}$

For admittances the reverse is true, that is

${\displaystyle Y_{\text{TP}}=Y_{1}+Y_{2}+Y_{3}+...\,}$
${\displaystyle {\frac {1}{Y_{\text{TS}}}}={\frac {1}{Y_{1}}}+{\frac {1}{Y_{2}}}+{\frac {1}{Y_{3}}}+...\,}$

Dealing with the reciprocals, especially in complex numbers, is more time consuming and error-prone than using linear addition. In general therefore, most RF engineers work in the plane where the circuit topography supports linear addition. The following table gives the complex expressions for impedance (real and normalised) and admittance (real and normalised) for each of the three basic passive circuit elements: resistance, inductance and capacitance. Using just the characteristic impedance (or characteristic admittance) and test frequency an equivalent circuit can be found and vice versa.

Expressions for Impedance and Admittance
Normalised by Impedance Z0 or Admittance Y0
Element TypeImpedance (Z or z) or Reactance (X or x)Admittance (Y or y) or Susceptance (B or b)
Real (${\displaystyle \Omega \,}$)Normalised (No Unit)Real (S)Normalised (No Unit)
Resistance (R)${\displaystyle Z=R\,}$${\displaystyle z={\frac {R}{Z_{0}}}=RY_{0}\,}$${\displaystyle Y=G={\frac {1}{R}}\,}$${\displaystyle y=g={\frac {1}{RY_{0}}}={\frac {Z_{0}}{R}}\,}$
Inductance (L)${\displaystyle Z=jX_{\text{L}}=j\omega L\,}$${\displaystyle z=jx_{\text{L}}=j{\frac {\omega L}{Z_{0}}}=j\omega LY_{0}\,}$${\displaystyle Y=-jB_{\text{L}}={\frac {-j}{\omega L}}}$${\displaystyle y=-jb_{\text{L}}={\frac {-j}{\omega LY_{0}}}={\frac {-jZ_{0}}{\omega L}}\,}$
Capacitance (C)${\displaystyle Z=-jX_{\text{C}}={\frac {-j}{\omega C}}\,}$${\displaystyle z=-jx_{\text{C}}={\frac {-j}{\omega CZ_{0}}}={\frac {-jY_{0}}{\omega C}}\,}$${\displaystyle Y=jB_{\text{C}}=j\omega C\,}$${\displaystyle y=jb_{\text{C}}=j{\frac {\omega C}{Y_{0}}}=j\omega CZ_{0}\,}$

## Using the Smith chart to solve conjugate matching problems with distributed components

Distributed matching becomes feasible and is sometimes required when the physical size of the matching components is more than about 5% of a wavelength at the operating frequency. Here the electrical behaviour of many lumped components becomes rather unpredictable. This occurs in microwave circuits and when high power requires large components in shortwave, FM and TV Broadcasting,

For distributed components the effects on reflection coefficient and impedance of moving along the transmission line must be allowed for using the outer circumferential scale of the Smith chart which is calibrated in wavelengths.

The following example shows how a transmission line, terminated with an arbitrary load, may be matched at one frequency either with a series or parallel reactive component in each case connected at precise positions.

Supposing a loss-free air-spaced transmission line of characteristic impedance ${\displaystyle Z_{0}=50\ \Omega }$, operating at a frequency of 800 MHz, is terminated with a circuit comprising a 17.5 ${\displaystyle \Omega }$ resistor in series with a 6.5 nanohenry (6.5 nH) inductor. How may the line be matched?

From the table above, the reactance of the inductor forming part of the termination at 800 MHz is

${\displaystyle Z_{L}=j\omega L=j2\pi fL=j32.7\ \Omega \,}$

so the impedance of the combination (${\displaystyle Z_{T}}$) is given by

${\displaystyle Z_{T}=17.5+j32.7\ \Omega \,}$

and the normalised impedance (${\displaystyle z_{T}}$) is

${\displaystyle z_{T}={\frac {Z_{T}}{Z_{0}}}=0.35+j0.65\,}$

This is plotted on the Z Smith chart at point P20. The line OP20 is extended through to the wavelength scale where it intersects at the point ${\displaystyle L_{1}=0.098\lambda \,}$. As the transmission line is loss free, a circle centred at the centre of the Smith chart is drawn through the point P20 to represent the path of the constant magnitude reflection coefficient due to the termination. At point P21 the circle intersects with the unity circle of constant normalised resistance at

${\displaystyle z_{P21}=1.00+j1.52\,}$.

The extension of the line OP21 intersects the wavelength scale at ${\displaystyle L_{2}=0.177\lambda \,}$, therefore the distance from the termination to this point on the line is given by

${\displaystyle L_{2}-L_{1}=0.177\lambda -0.098\lambda =0.079\lambda \,}$

Since the transmission line is air-spaced, the wavelength at 800 MHz in the line is the same as that in free space and is given by

${\displaystyle \lambda ={\frac {c}{f}}\,}$

where ${\displaystyle c\,}$ is the velocity of electromagnetic radiation in free space and ${\displaystyle f\,}$ is the frequency in hertz. The result gives ${\displaystyle \lambda =375\ \mathrm {mm} \,}$, making the position of the matching component 29.6 mm from the load.

The conjugate match for the impedance at P21 (${\displaystyle z_{match}\,}$) is

${\displaystyle z_{match}=-j(1.52),\!}$

As the Smith chart is still in the normalised impedance plane, from the table above a series capacitor ${\displaystyle C_{m}\,}$ is required where

${\displaystyle z_{match}=-j1.52={\frac {-j}{\omega C_{m}Z_{0}}}={\frac {-j}{2\pi fC_{m}Z_{0}}}\,}$

Rearranging, we obtain

${\displaystyle C_{m}={\frac {1}{(1.52)\omega Z_{0}}}={\frac {1}{(1.52)(2\pi f)Z_{0}}}}$.

Substitution of known values gives

${\displaystyle C_{m}=2.6\ \mathrm {pF} \,}$

To match the termination at 800 MHz, a series capacitor of 2.6 pF must be placed in series with the transmission line at a distance of 29.6 mm from the termination.

An alternative shunt match could be calculated after performing a Smith chart transformation from normalised impedance to normalised admittance. Point Q20 is the equivalent of P20 but expressed as a normalised admittance. Reading from the Smith chart scaling, remembering that this is now a normalised admittance gives

${\displaystyle y_{Q20}=0.65-j1.20\,}$

(In fact this value is not actually used). However, the extension of the line OQ20 through to the wavelength scale gives ${\displaystyle L_{3}=0.152\lambda \,}$. The earliest point at which a shunt conjugate match could be introduced, moving towards the generator, would be at Q21, the same position as the previous P21, but this time representing a normalised admittance given by

${\displaystyle y_{Q21}=1.00+j1.52\,}$.

The distance along the transmission line is in this case

${\displaystyle L_{2}+L_{3}=0.177\lambda +0.152\lambda =0.329\lambda \,}$

which converts to 123 mm.

The conjugate matching component is required to have a normalised admittance (${\displaystyle y_{match}}$) of

${\displaystyle y_{match}=-j1.52\,}$.

From the table it can be seen that a negative admittance would require an inductor, connected in parallel with the transmission line. If its value is ${\displaystyle L_{m}\,}$, then

${\displaystyle -j1.52={\frac {-j}{\omega L_{m}Y_{0}}}={\frac {-jZ_{0}}{2\pi fL_{m}}}\,}$

This gives the result

${\displaystyle L_{m}=6.5\ \mathrm {nH} \,}$

A suitable inductive shunt matching would therefore be a 6.5 nH inductor in parallel with the line positioned at 123 mm from the load.

## Using the Smith chart to analyze lumped element circuits

The analysis of lumped element components assumes that the wavelength at the frequency of operation is much greater than the dimensions of the components themselves. The Smith chart may be used to analyze such circuits in which case the movements around the chart are generated by the (normalized) impedances and admittances of the components at the frequency of operation. In this case the wavelength scaling on the Smith chart circumference is not used. The following circuit will be analyzed using a Smith chart at an operating frequency of 100 MHz. At this frequency the free space wavelength is 3 m. The component dimensions themselves will be in the order of millimetres so the assumption of lumped components will be valid. Despite there being no transmission line as such, a system impedance must still be defined to enable normalization and de-normalization calculations and ${\displaystyle Z_{0}=50\ \Omega \,}$ is a good choice here as ${\displaystyle R_{1}=50\ \Omega \,}$. If there were very different values of resistance present a value closer to these might be a better choice.

The analysis starts with a Z Smith chart looking into R1 only with no other components present. As ${\displaystyle R_{1}=50\ \Omega \,}$ is the same as the system impedance, this is represented by a point at the centre of the Smith chart. The first transformation is OP1 along the line of constant normalized resistance in this case the addition of a normalized reactance of -j0.80, corresponding to a series capacitor of 40 pF. Points with suffix P are in the Z plane and points with suffix Q are in the Y plane. Therefore, transformations P1 to Q1 and P3 to Q3 are from the Z Smith chart to the Y Smith chart and transformation Q2 to P2 is from the Y Smith chart to the Z Smith chart. The following table shows the steps taken to work through the remaining components and transformations, returning eventually back to the centre of the Smith chart and a perfect 50 ohm match.

Smith chart steps for analysing a lumped element circuit
TransformationPlanex or y Normalized ValueCapacitance/InductanceFormula to SolveResult
${\displaystyle O\rightarrow P_{1}\,}$${\displaystyle Z\,}$${\displaystyle -j0.80\,}$Capacitance (Series)${\displaystyle -j0.80={\frac {-j}{\omega C_{1}Z_{0}}}\,}$${\displaystyle C_{1}=40\ \mathrm {pF} \,}$
${\displaystyle Q_{1}\rightarrow Q_{2}\,}$${\displaystyle Y\,}$${\displaystyle -j1.49\,}$Inductance (Shunt)${\displaystyle -j1.49={\frac {-j}{\omega L_{1}Y_{0}}}\,}$${\displaystyle L_{1}=53\ \mathrm {nH} \,}$
${\displaystyle P_{2}\rightarrow P_{3}\,}$Z${\displaystyle -j0.23\,}$Capacitance (Series)${\displaystyle -j0.23={\frac {-j}{\omega C_{2}Z_{0}}}\,}$${\displaystyle C_{2}=138\ \mathrm {pF} \,}$
${\displaystyle Q_{3}\rightarrow O\,}$Y${\displaystyle +j1.14\,}$Capacitance (Shunt)${\displaystyle +j1.14={\frac {j\omega C_{3}}{Y_{0}}}\,}$${\displaystyle C_{3}=36\ \mathrm {pF} \,}$

## 3D Smith chart

A generalized 3D Smith chart based on the extended complex plane (Riemann sphere) and inversive geometry was proposed in 2011. The chart unifies the passive and active circuit design on little and big circles on the surface of a unit sphere using the stereographic conformal map of the reflection coefficient's generalized plane. Considering the point at infinity, the space of the new chart includes all possible loads. The north pole is the perfect matching point, while the south pole is the perfect mismatch point. [10]

## Related Research Articles

The characteristic impedance or surge impedance (usually written Z0) of a uniform transmission line is the ratio of the amplitudes of voltage and current of a single wave propagating along the line; that is, a wave travelling in one direction in the absence of reflections in the other direction. Alternatively and equivalently it can be defined as the input impedance of a transmission line when its length is infinite. Characteristic impedance is determined by the geometry and materials of the transmission line and, for a uniform line, is not dependent on its length. The SI unit of characteristic impedance is the ohm.

The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the change per unit length, but it is otherwise dimensionless. In the context of two-port networks and their cascades, propagation constant measures the change undergone by the source quantity as it propagates from one port to the next.

In radio engineering and telecommunications, standing wave ratio (SWR) is a measure of impedance matching of loads to the characteristic impedance of a transmission line or waveguide. Impedance mismatches result in standing waves along the transmission line, and SWR is defined as the ratio of the partial standing wave's amplitude at an antinode (maximum) to the amplitude at a node (minimum) along the line.

In radio-frequency engineering, a transmission line is a specialized cable or other structure designed to conduct alternating current of radio frequency, that is, currents with a frequency high enough that their wave nature must be taken into account. Transmission lines are used for purposes such as connecting radio transmitters and receivers with their antennas, distributing cable television signals, trunklines routing calls between telephone switching centres, computer network connections and high speed computer data buses.

A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting expansion to one dimension or two. There is a similar effect in water waves constrained within a canal, or guns that have barrels which restrict hot gas expansion to maximize energy transfer to their bullets. Without the physical constraint of a waveguide, wave amplitudes decrease according to the inverse square law as they expand into three dimensional space.

In electrical engineering, admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the reciprocal of impedance. The SI unit of admittance is the siemens ; the older, synonymous unit is mho, and its symbol is ℧. Oliver Heaviside coined the term admittance in December 1887.

In electronics, impedance matching is the practice of designing the input impedance of an electrical load or the output impedance of its corresponding signal source to maximize the power transfer or minimize signal reflection from the load.

The Heaviside condition, named for Oliver Heaviside (1850–1925), is the condition an electrical transmission line must meet in order for there to be no distortion of a transmitted signal. Also known as the distortionless condition, it can be used to improve the performance of a transmission line by adding loading to the cable.

The transmission coefficient is used in physics and electrical engineering when wave propagation in a medium containing discontinuities is considered. A transmission coefficient describes the amplitude, intensity, or total power of a transmitted wave relative to an incident wave.

The Π pad is a specific type of attenuator circuit in electronics whereby the topology of the circuit is formed in the shape of the Greek letter "Π".

Image impedance is a concept used in electronic network design and analysis and most especially in filter design. The term image impedance applies to the impedance seen looking into a port of a network. Usually a two-port network is implied but the concept can be extended to networks with more than two ports. The definition of image impedance for a two-port network is the impedance, Zi 1, seen looking into port 1 when port 2 is terminated with the image impedance, Zi 2, for port 2. In general, the image impedances of ports 1 and 2 will not be equal unless the network is symmetrical with respect to the ports.

Constant k filters, also k-type filters, are a type of electronic filter designed using the image method. They are the original and simplest filters produced by this methodology and consist of a ladder network of identical sections of passive components. Historically, they are the first filters that could approach the ideal filter frequency response to within any prescribed limit with the addition of a sufficient number of sections. However, they are rarely considered for a modern design, the principles behind them having been superseded by other methodologies which are more accurate in their prediction of filter response.

m-derived filters or m-type filters are a type of electronic filter designed using the image method. They were invented by Otto Zobel in the early 1920s. This filter type was originally intended for use with telephone multiplexing and was an improvement on the existing constant k type filter. The main problem being addressed was the need to achieve a better match of the filter into the terminating impedances. In general, all filters designed by the image method fail to give an exact match, but the m-type filter is a big improvement with suitable choice of the parameter m. The m-type filter section has a further advantage in that there is a rapid transition from the cut-off frequency of the pass band to a pole of attenuation just inside the stop band. Despite these advantages, there is a drawback with m-type filters; at frequencies past the pole of attenuation, the response starts to rise again, and m-types have poor stop band rejection. For this reason, filters designed using m-type sections are often designed as composite filters with a mixture of k-type and m-type sections and different values of m at different points to get the optimum performance from both types.

A quarter-wave impedance transformer, often written as λ/4 impedance transformer, is a transmission line or waveguide used in electrical engineering of length one-quarter wavelength (λ), terminated with some known impedance. It presents at its input the dual of the impedance with which it is terminated.

A signal travelling along an electrical transmission line will be partly, or wholly, reflected back in the opposite direction when the travelling signal encounters a discontinuity in the characteristic impedance of the line, or if the far end of the line is not terminated in its characteristic impedance. This can happen, for instance, if two lengths of dissimilar transmission lines are joined together.

The primary line constants are parameters that describe the characteristics of conductive transmission lines, such as pairs of copper wires, in terms of the physical electrical properties of the line. The primary line constants are only relevant to transmission lines and are to be contrasted with the secondary line constants, which can be derived from them, and are more generally applicable. The secondary line constants can be used, for instance, to compare the characteristics of a waveguide to a copper line, whereas the primary constants have no meaning for a waveguide.

Metal-mesh optical filters are optical filters made from stacks of metal meshes and dielectric. They are used as part of an optical path to filter the incoming light to allow frequencies of interest to pass while reflecting other frequencies of light.

The T pad is a specific type of attenuator circuit in electronics whereby the topology of the circuit is formed in the shape of the letter "T".

A frequency-selective surface (FSS) is any thin, repetitive surface designed to reflect, transmit or absorb electromagnetic fields based on the frequency of the field. In this sense, an FSS is a type of optical filter or metal-mesh optical filters in which the filtering is accomplished by virtue of the regular, periodic pattern on the surface of the FSS. Though not explicitly mentioned in the name, FSS's also have properties which vary with incidence angle and polarization as well - these are unavoidable consequences of the way in which FSS's are constructed. Frequency-selective surfaces have been most commonly used in the radio frequency region of the electromagnetic spectrum and find use in applications as diverse as the aforementioned microwave oven, antenna radomes and modern metamaterials. Sometimes frequency selective surfaces are referred to simply as periodic surfaces and are a 2-dimensional analog of the new periodic volumes known as photonic crystals.

## References

1. Smith, P. H.; Transmission Line Calculator; Electronics, Vol. 12, No. 1, pp 29-31, January 1939
2. Smith, P. H.; An Improved Transmission Line Calculator; Electronics, Vol. 17, No. 1, p 130, January 1944
3. Ramo, Whinnery and Van Duzer (1965); "Fields and Waves in Communications Electronics"; John Wiley & Sons; pp 35-39. ISBN
4. Pozar, David M. (2005); Microwave Engineering, Third Edition (Intl. Ed.); John Wiley & Sons, Inc.; pp 64-71. ISBN   0-471-44878-8.
5. Gonzalez, Guillermo (1997); Microwave Transistor Amplifiers Analysis and Design, Second Edition; Prentice Hall NJ; pp 93-103. ISBN   0-13-254335-4.
6. Gonzalez, Guillermo (1997) (op. cit);pp 98-101
7. Gonzalez, Guillermo (1997) (op. cit);p 97
8. Hayt, William H Jr.; "Engineering Electromagnetics" Fourth Ed; McGraw-Hill International Book Company; pp 428–433. ISBN   0-07-027395-2.
9. Davidson, C. W.;"Transmission Lines for Communications with CAD Programs";Macmillan; pp 80-85. ISBN   0-333-47398-1
10. Andrei Muller, Pablo Soto, D. Dascalu, D. Neculoiu, V.E. Boria A 3D Smith chart based on the Riemann sphere for Active and Passive Microwave Circuits, Microwave and Wireless Components Letters doi : 10.1109/LMWC.2011.2132697
• Mizuhashi, T., Theory of four-terminal impedance transformation circuit and matching circuit, The Journal of the Institute of Electrical Communication Engineers of Japan, pp. 1053–1058, December 1937.
• P.H.Smith 1969 Electronic Applications of the Smith Chart. Kay Electric Company
• For an early representation of this graphical depiction before they were called 'Smith Charts', see G. A. Campbell, "Cisoidal Oscillations", Proc. AIEE, 30, 1-6, pp. 789–824 (1911). In particular, Fig. 13 on p. 810.